# Extra dimensions don't go with dark energy (new Steinhardt no-go theorems)

1. Nov 11, 2008

### marcus

This applies to the small extra dimensions. The rolled-up compact kind that we wouldn't see if they were there. Paul Steinhardt has just posted a paper proving no-go theorems that exhibit an impressive degree of incompatibility between compact extra dimensions on the one hand versus the accelerated expansion we observe today, and also versus the widely accepted concept of early universe inflation.

The extent of incompatibility is surprising, so this paper is likely to have repercussions and could arouse controversy. It appeared today. Here's the abstract:

http://arxiv.org/abs/0811.1614
Dark Energy, Inflation and Extra Dimensions
Paul J. Steinhardt, Daniel Wesley
26 pages, 1 figure
(Submitted on 11 Nov 2008)

"We consider how accelerated expansion, whether due to inflation or dark energy, imposes strong constraints on fundamental theories obtained by compactification from higher dimensions. For theories that obey the null energy condition (NEC), we find that inflationary cosmology is impossible for a wide range of compactifications; and a dark energy phase consistent with observations is only possible if both Newton's gravitational constant and the dark energy equation-of-state vary with time. If the theory violates the NEC, inflation and dark energy are only possible if the NEC-violating elements are inhomogeneously distributed in the compact dimensions and vary with time in precise synchrony with the matter and energy density in the non-compact dimensions. Although our proofs are derived assuming general relativity applies in both four and higher dimensions and certain forms of metrics, we argue that similar constraints must apply for more general compactifications."

Last edited: Nov 11, 2008
2. Nov 11, 2008

### Haelfix

The null energy condition is a very strong constraint on model building in general. Violating it is very dangerous, but is often done in quantum cosmology phenomenology (many of the oft cited models of the last 10 years violate it in some way).

There are some technical issues with the paper in question and some roundabout ways out of the conclusions. The boundary condition assumptions in particular are very strong.

3. Nov 11, 2008

### ensabah6

If the paper survives peer review and scrutiny, would it be said, then, that some classes of string theories that rely on certain compactifications (esp re: Yau-Calibi manifolds) hiding 6 or 7 higher dimensions, leaving 4 large dimensions, nonetheless are inconsistent with certain observations of dark energy (positive cosmological constant)?

4. Nov 12, 2008

### Coin

Aren't most string theorists moving toward large-extra-dimension "braneworld" models anyway?

5. Nov 12, 2008

### Demystifier

I wouldn't say so.
The landscape paradigm, which, among other things, is supposed to solve the cosmological constant problem, is mainly based on small extra dimensions.

6. Nov 12, 2008

### Haelfix

"If the paper survives peer review and scrutiny, would it be said, then, that some classes of string theories that rely on certain compactifications (esp re: Yau-Calibi manifolds) hiding 6 or 7 higher dimensions, leaving 4 large dimensions, nonetheless are inconsistent with certain observations of dark energy (positive cosmological constant)? "

Doubtful, what it does do is constrain the moduli dynamics, the initial conditions or begs for NEC violation (which probably already might appear, at least microscopically. Globally its hard to make it physical). I'd say its more of an issue for vanilla Randall-Sundrum where it adds a lot of finetuning.

7. Nov 12, 2008

### p-brane

The elephant in the room here is our woefully incomplete understanding of the gravitational physics of the vacuum which remains very nearly a complete mystery. It may indeed be that the true nature of dark energy is something revolutionary and either renders these no go theorems meaningless or evades them. More generally, theres just so much wiggle room here that I dont really think people are going to lose much sleep over this paper.

8. Nov 15, 2008

### marcus

It might be interesting to examine the paper in enough detail to distinguish the different no-go theorems for which Steinhardt and Wesley offer proofs, and to rephrase in one's own words what it is they say they have shown.

As a convenience to anyone who wants to do that, I will list the no-go theorems. In the paper they are interspersed with discussion and particular ones you might be looking for are not always easy to find. So I will give page references, in case you want to find the discussion surrounding a particular theorem. Here they are gathered together:

==excerpts from Steinhardt Wesley==
(Theorems labeled IA, IB, etc. refer to models obeying NEC and models labeled IIA,IIB, etc. refer to models that violate NEC...
In the no-go theorems presented in this paper, the term “compactified models” is short for
models satisfying the GR, flatness, boundedness and metric conditions discussed in Sec. IIA.)

Dark Energy No-go Theorem IA: LamdaCDM (the current concordance model in cosmology) is incompatible with compactified models satisfying the NEC. (page 10)

Dark Energy No-go Theorem IB: Dark energy models with constant wDE less than wtransient or time-varying wDE whose value remain less than wtransient for a continuous period lasting more than a few Hubble times are incompatible with compactified models satisfying the NEC. (page 11)

Inflationary No-go Theorem IA: Inflationary models consistent with observations are incompatible with compactified models satisfying the NEC. (bottom of page 11)

Inflationary Corollary: Compactified models satisfying the NEC are counterexamples to the common assertion that inflation with nearly scale-invariant spectra are an inevitable consequence given chaotic or generic initial conditions after the big bang. (page 12)

Dark Energy No-go Theorem IC: All dark energy models are incompatible with compactified models satisfying the NEC if the moduli fields are frozen (or, specifically, GN is constant, in the case of CRF models). (page 13)

Inflationary/Dark Energy No-go Theorem IIA: Inflation and dark energy are incompatible with compactified models (with fixed moduli) if the NEC is satisfied in the compact dimensions (i.e., ρ + pk ≥ 0 for all t and ym) — whether or not NEC is violated in the non-compact directions. (page 14)

Inflationary/Dark Energy No-go Theorem IIB: Inflation and dark energy are incompatible with compactified models (with fixed moduli) for which the net NEC violation (ρ + pk) is time-independent. (page 14)

Inflationary Corollary: Inflationary cosmology is only compatible with compactified theories that include an NEC violating component in the compact dimensions whosemagnitude is of order the vacuum density...<snip>...; such that <ρ + pk>A* switches from positive to negative when inflation begins and switches back when inflation is complete. (bottom page 14, the notation employs a concept of A-averaging introduced by Wesley in http://arxiv.org/abs/0802.3214)

Inflationary/Dark Energy No-go Theorem IIC: Inflation and dark energy are incompatible with compactified models with fixed moduli if the warp factor Omega(t, y) is non-trivial and has continuous first derivative and if any of the following quantities is homogeneous in y:
1. ρ + p3;
2. xρ + pk for RF metric, for any (1/2)(1 − 3w) > x > 4(k − 1)/3k;
3. ρ for CRF metric for k > 4;
4. 2ρ + pk for CRF metric for k > 3 and w > −1.
(page 15)

Inflationary/Dark Energy No-go Theorem IID: Inflation and dark energy are incompatible with compactified models with fixed moduli if the warp factor Omega(t, y) is non-trivial if ρ + pk is homogeneous. (page 16)

Inflationary/Dark Energy No-go Theorem IIE: Inflation and dark energy are incompatible with compactified models with fixed moduli if wk(A∗) ≡ <pk>A* /<ρk>A* > −1 for <pk>A* > 0 or if wk(A∗) ≡ <pk>A* /<ρk>A* < −1 for <pk>A* < 0 .
(page 17, again this theorem's statement uses the concept of A-averaging introduced by Wesley in http://arxiv.org/abs/0802.3214)

Last edited: Nov 15, 2008
9. Nov 15, 2008

### marcus

I think it's important not just to respond to what you think Steinhardt and Wesley might be saying, after a perhaps cursory inspection, but to try to understand their actual conclusions--that they draw at the end after proving this remarkable batch of No-Go theorems.

===sample excerpts, pages 20 and 21===
The essence of this paper is that cosmic acceleration is surprisingly difficult to incorporate in compactified models. The problem arises in trying to satisfy simultaneously the 4d and higher dimensional Einstein equations. Both must be satisfied for any equation-of-state, but we have shown that, for the metrics assumed in this paper, this requires increasingly exotic conditions as the universe goes from decelerated to accelerated expansion or, equivalently, as w decreases below −1/3. For dark energy models, either moduli fields (including GN) have to change with time at a rate that is nearly ruled out (and may soon be excluded observationally altogether[24]) or NEC must be violated. For inflation, only the second option remains viable.

If the NEC is violated, it must be violated in the compact dimensions; it must be violated strongly (wk significantly below the minimally requisite value for NEC violation); and the violation in the compact dimensions must vary with time in a manner that precisely tracks the equation-of-state in the 4d effective theory. For example, in realistic cosmological models, there are known matter and radiation components (baryons and photons, for example) that contribute to the energy and density of the 4d effective theory but are not normally related to NEC violation. Nevertheless, the no-go theorems say that the magnitude of NEC violation must vary with time in sync with how the conventional matter and radiation energy density and pressure evolve.

Satisfying these equations for LamdaCDM is difficult, but satisfying them for inflation is even harder...
==endquote==

This indicates that they are considering both cases: where the NEC is not violated and where it is violated. In both cases they find that compact dimension models must apparently go thru some rather elaborate and exotic contortions to avoid being ruled out. How are we to take this?

We all know that Steinhardt has been supportive of stringy speculation for much of his career. What he is best-known for, braneclash cosmology (ekpyrotic, cyclic etc), was specifically motivated as a safe haven for superstring. Already back in the 1990s Steinhardt suspected inflation was incompatible with compact extra dimensions, so he and Turok came up with a string-friendly cosmology that did not need inflation.

In sum, he has known about these indications of multiple-allergy incompatibilities of various kinds, and has had them on his mind for over 10 years. But now he says (I believe quite frankly) that he is surprised. Surprised, I take it, by the fact that the signs of tissue-rejection are so severe, and extend also to the dark energy acceleration of the standard LambdaCDM consensus model.
It is not just the presumed episode of inflation, it is also the much milder accelerated expansion we see now, that string is out of joint with.

What else might he be surprised about? I guess by the generality and comprehensiveness of the apparent incompatibility. By and large, the No-Go theorems do not pick out some particular detail of a particular theory. They apply to broad classes of extra-dimension models. Both NEC-violating and NEC-obeying alike.

So what's the overall message, to take away from this. Well Steinhardt has always been supportive of string research (although apparently not fond of the Anthropic Landscape) and he doesn't claim to completely invalidate stringy thinking here either. I think the message is rather one of friendly caution which some readers may decide to heed, and others to ignore. He doesn't attack, or speak as a critic: he says to be wary---he points to signs of trouble.

==quote page 22==
In general, the no-go theorems are powerful because they span a broad sweep of theories.
They say that one should be wary of focusing on one localized region of the extra-dimensions,... since there are non-trivial global constraints.
==endquote==

What kind of impact this paper has, and how it influences the future course of research, will of course depend on how well the proofs stand up under scrutiny. If any reputable physicist finds fault or has serious reservations, I'd expect to see a Comments paper posted on arxiv. Steinhardt is important enough that his paper is not the kind of thing I'd expect the community to squabble about in blog, or to handle at the handwave and rumor level. But we'll see. Any type of reaction will be fun!

Last edited: Nov 15, 2008
10. Nov 16, 2008

### Haelfix

Violating energy conditions is a big subject matter with a lot of research and still poorly understood.

For instance, very often violating either the dominant energy condition or the NEC will have a tendency to create Cauchy horizons and/or CTCs. At the quantum level, you will often end up with really bad vacuum instabilities in highly boosted frames (tachyons/naked ghosts etc)

Unfortunately many prominent cosmologies including some phenomenologically attractive ones (cyclic universe, ekyprotic, several versions of inflation, string gas, bouncing solutions) tend to violate the NEC at some level. So a lot of the research is in figuring out when such and such a process is unphysical and when its ok (eg maybe its ok to violate it microscopically but it needs to hold on average macroscopically)

At some level (say quantum gravity), they might be ok, but the vacuum as mentioned is still not understood correctly and it makes extrapolation into semiclassical regimes highly tentative, which is why these nogo theorems are likely irrelevant.

Other than that, in this particular paper, the assumptions are extremely strong and there is still a lot of wiggle room with playing with the moduli and initial conditions as I mentioned previously

11. Nov 16, 2008

### marcus

I'd say we don't really know about the likely relevance or irrelevance at this point. A good leading indicator would be how defensive the more articulate string people are. If Lubos Motl launches a diatribe tearing the paper down, then I should think that would be a sign that the No-Go theorems of this sort are likely to play an important role.

If Jacques Distler tries to dismiss or minimize the Steinhardt Wesley paper that would be a great sign that the theorems indeed pose a serious challenge, wouldn't you say?

In the meanwhile, without any meaningful signals as to the reaction, I checked to see who Wesley is (Steinhardt is highly prominent, no need to give detail on that.)

http://en.scientificcommons.org/daniel_h_wesley
At Cambridge he seems to have worked with Turok, who is now director of Perimeter Institute.
At Princeton he evidently worked with Steinhardt.
Here is his CV
http://www.damtp.cam.ac.uk/user/dhw22/cv/index.html
There is much more to the CV than what I quote---the list of awards and honors is impressive---but here is the barebones version:
* Postdoctoral researcher, CTC/DAMTP Cambridge University, 2006-present

* Ph. D., Physics, Princeton University, 2006.

* M.S., Physics, Princeton University, 2003.

* Certificate of Advanced Study in Mathematics (Mathematical Tripos/Part III), awarded with Distinction, Cambridge University, 2001.

* A.B., Physics, Summa cum Laude, Princeton University, 2000.

I see that Steinhardt and Wesley indicate have another paper in preparation, in which they prove theorems which extend the results in this one.

12. Nov 16, 2008

### friend

This subject confuses me a little. I understand String theory to be background dependent, meaning that the background metric is assumed to be flat. So it seems as though String theory has nothing to say about the background metric, whether it be curved, flat, or expanding. So how can these no-go theorems be valid in String theory which generally does not address the background metric? Thanks.

13. Nov 16, 2008

### marcus

String thinkers mean something else when they say background dependent/independent.
Not sure what. In the nonstring QG community a theory is called B.D. if it is defined using a fixed spacetime geometry---even if the geometry isn't flat. I don't think that's an important point here, just wanted to be clear about it.

So much noise surrounds the term B.I. that it can be easier to simply say what you mean in more words, like "doesn't need to start with a metric on the manifold, doesn't commit to a particular geometry." In General Relativity the geometry is arrived at dynamically in interaction with matter, at no point is it put in by hand. The nonstring QG people were using the term B.I. as a shorthand phrase to point to this feature which GR has in an exemplary way, and which they wanted their quantized GR to also have.

So in the Nonstring QG sense, a theory is B.D. if at some point in defining it you have to choose a geometry to specify (even if the choice you have is very wide, the point is that you specify what the background looks like).
==============================

About your question. Expansion is a General Relativity thing. The 4D Minkowski space which we call flat does not expand. As soon as you start talking inflation or dark energy accelerated expansion, you are talking GR.
GR can be defined in any number of dimensions. In particular GR can be defined on manifolds with rolled up extra dimensions. Various stringy schemes need these manifolds.

The catch is, that we know have to have these manifolds with their compactified extra dimensions expand according to GR and they apparently don't get along with GR very well if you expect them to undergo inflation scenarios and stuff like that.

Last edited: Nov 16, 2008
14. Nov 16, 2008

### p-brane

Another point worth making is that no entirely satisfactory inflationary scenario has yet been found. So if inflation, or how we currently think about it, is somehow wrong, than one shouldn't be surprised if inflation, or how we currently understand it, cannot be embedded in the correct theory. Perhaps it's time to address the fundamental difficulties associated with conventional inflationary scenarios.

Last edited: Nov 16, 2008
15. Nov 16, 2008

### marcus

p-brane
you say, "Another point worth making is that no entirely satisfactory inflationary scenario has yet been found..." You might want to expand your statement to cover presentday accelerated expansion. Because about half of their No-Go theorems apply not to early-universe inflation but to the accelerated expansion first observed in 1998, which is so much studied these days.

It is THIS that strikes me as the primary concern---the seeming incompatibility of extra dimensions with what we actually observe happening in the universe today. Perhaps you have some words you would like to say about this as well?

friend,
of course it's possible to quibble about the right general statement to make about these indications of incompatibility. But you can see for yourself by looking at the proofs of these 10 or eleven theorems quoted earlier, that again and again what Wesley and Steinhardt are finding is that it is very difficult to synchronize the accelerated expansion of the extra dimensions with the accelerated expansion of the ordinary dimensions. It is the coordinated expansion that doesn't seem to want to happen naturally.

But rather than rely on my generalized overview of it, or anyone else's, it would be better to actually have a look at how the theorems are proved in the paper and see what kind of things the proofs are based on. I'll relist the theorems for handy reference, but for the proofs you have to download the PDF:

Last edited: Nov 16, 2008
16. Nov 16, 2008

### p-brane

A simple example of how our ignorance about the true nature of the gravitational vacuum is that its quite possible that dark energy isnt actually an energy - at least not as we currently understand it - so that the NEC would not apply to it.

17. Dec 2, 2008

### marcus

This is potentially quite an important paper, and one for which Steinhardt and Wesley have a follow-up paper in preparation. I'll refresh our memory of what it's about.

They prove 11 no-go theorems and corollaries applicable to various different string-theoretical schemes. These show incompatibility of three sorts.

A. Incompatibility with the standard LCDM cosmology model---that is with the kind of gradually accelerated expansion astronomers believe they are observing

B. Incompatibility with inflation scenarios---the brief episode of rapidly accelerated expansion postulated to have occurred in early universe.

C. Incompatibility with both standard LCDM cosmology and with inflation

Of the 11 results, three show incompatibility with LCDM (type A),
three others show incompatibility with inflation (type B), and
five more show incompatibility with both (type C)

In all cases the incompatibility is due to the "rolled up" extra dimensions---to put it simply, the "compactified" extra dimensions don't stay rolled up properly during accelerated expansion. Again to simplify: unless physical constants are varied with time in exactly the right fashion, the correct balance of moduli is destroyed, the extra dimensions become either too rolled up or not rolled up enough, and the universe goes to pot.

Here's a quote from pages 20 and 21 that kind of sums it up:
==quote==
The essence of this paper is that cosmic acceleration is surprisingly difficult to incorporate in compactified models. The problem arises in trying to satisfy simultaneously the 4d and higher dimensional Einstein equations. Both must be satisfied for any equation-of-state, but we have shown that, for the metrics assumed in this paper, this requires increasingly exotic conditions as the universe goes from decelerated to accelerated expansion or, equivalently, as w decreases below −1/3. For dark energy models, either moduli fields (including GN) have to change with time at a rate that is nearly ruled out (and may soon be excluded observationally altogether[24]) or NEC must be violated. For inflation, only the second option remains viable.

If the NEC is violated, it must be violated in the compact dimensions; it must be violated strongly (wk significantly below the minimally requisite value for NEC violation); and the violation in the compact dimensions must vary with time in a manner that precisely tracks the equation-of-state in the 4d effective theory. For example, in realistic cosmological models, there are known matter and radiation components (baryons and photons, for example) that contribute to the energy and density of the 4d effective theory but are not normally related to NEC violation. Nevertheless, the no-go theorems say that the magnitude of NEC violation must vary with time in sync with how the conventional matter and radiation energy density and pressure evolve.

Satisfying these equations for LamdaCDM is difficult, but satisfying them for inflation is even harder...
==endquote==

Some people may have given the paper a quick reading and failed to notice that the no-go theorems cover both cases where the Null Energy Condition is obeyed and cases where it is violated. It would be a mistake to suppose that the NEC is the decisive issue here. The main point of vulnerability which the theorems focus on is the simple and general one of needing compactified extra dimensions.

Just as a side comment, the use of compactified extra dimensions is also at the root of the Landscape problem----the huge number of distinct vacua, each with its own version of physics, occurs because of the great variety of ways that extra dimensions can be rolled up and packaged.

Probabaly we should go through the paper and gather up a list of different compact extra dimension models affected by the various no-go theorems and corollaries. Also keep a look out for the follow-up paper by the same authors, as it may cover further cases.

Last edited: Dec 2, 2008
18. Dec 2, 2008

### atyy

Wow, did they get rid of the landscape? Does that make string theory viable?

One thought, I saw you posted a paper about the LCDM model being incompatible at 2 sigma with some data. If that continues to hold, would that be an out from the Steinhardt and Wesley paper?

19. Dec 2, 2008

### marcus

As a mere interested observer, all I can say is that it would be sublimely wonderful if the string people could find a way to do everything they want to without compactified extra dimensions!

All Steinhardt and Wesley are able to do is show that extra dimensions are apt to cause trouble.

And the KKLT paper of 2003 which began the Landscape rumpus also showed the same thing, that extra dimensions can cause unexpected difficulties.

But alas, no one I know of has seen how to get rid of the Landscape.

This is interesting, I want to reply but have to go to dinner. Will get back later or tomorrow.

20. Dec 17, 2008

### marcus

I don't think that particular paper holds out much hope that string approaches could prove viable. The essential thing it does is suggest a mild modification in w, the dark energy equation of state, or in the gravitational constant G--no reason for that to affect Steinhardt Wesley conclusions.

The Steinhardt Wesley No-Go results are robust in the sense that they depend only on the rough overall feature of LCDM, the simple fact of accelerated expansion, not on details like having w exactly equal -1. In the "Six puzzles" paper, LCDM or a slight adjustment where a couple of parameters are allowed to evolve over time still comes out looking better than alternatives.

That's my take on the paper that you referred to. It is an interesting, very readable paper which I'd invite anyone to take a look at. Here's a quote from conclusions. http://arxiv.org/pdf/0811.4684v1
====quote Perivolaropoulos "Six puzzles" paper=====

...It is interesting to attempt to identify universal features which connect these puzzles and could therefore provide a guide for their simultaneous resolution. The large scale coherent velocity ﬂows along with the high density dark matter haloes for both galaxies and clusters seem to hint towards a more eﬀective mechanism for structure formation at early times (z > 1) than implied by ΛCDM . This improved eﬀectiveness could possibly be provided by a mild evolution of Newton’s constant G (higher G at z > 0.5) or by an evolution of the dark energy equation of state w such that w(z ) > −1 at z > ∼ 0.5 [56].

Both of these eﬀects are expected to amplify structure formation at early times and it would be interesting to analyze quantitatively the predictions implied by the evolution of G or w with respect to the velocity ﬂow and high dark matter density puzzles. The Bright High z SnIa puzzle would also beneﬁt signiﬁcantly by a mild evolution of w or G which would imply stronger deceleration at z > 1 than implied by ΛCDM.

The improved eﬃciency of gravity at early times could also help emptying the voids from dark matter haloes and their corresponding galaxies thus making theoretical predictions more consistent with observations. On the other hand, the increased gravitational acceleration would also produce higher peculiar velocities that could lead to more mass inside the voids. Therefore, the predicted emptiness of voids in models with an evolving G or w requires a detailed study.
====endquote====

Last edited: Dec 17, 2008
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